few_shots: Example:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K
options: Interpret functions that arise in applications in terms of the context.

Extend the domain of trigonometric functions using the unit circle.

Interpret expressions for functions in terms of the situation they model.

Analyze functions using different representations.

Model periodic phenomena with trigonometric functions.

Build a function that models a relationship between two quantities.

Use functions to model relationships between quantities.

Construct and compare linear and exponential models and solve problems.

Prove and apply trigonometric identities.

Build new functions from existing functions.

Define, evaluate, and compare functions.

Understand the concept of a function and use function notation.
problem_activity: Activity
The pattern in this activity creates the Fibonacci sequence. One purpose of this task is for students to understand that sometimes it is straightforward to define a sequence recursively and difficult to determine a definition for the
\(n^{\text{th}}\)
term. This task also challenges students to expand their notion of recursive definitions since the current term in a Fibonacci sequence depends on the previous two terms rather than just the previous term.
This sequence was chosen to provide a contrast to the cake activity. With the cake, students had a situation whose domain does not reasonably extend above a certain value given physical constraints. Since this sequence is a mathematical construct, the domain has no upper bound restrictions on it.
Launch
Distribute 1 piece of graph paper to each student. Ask them to complete the first question and pause. Ensure that they are interpreting the description of the pattern correctly before they proceed with the rest of the activity.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention
Student Facing
On graph paper, draw a square of side length 1. Draw another square of side length 1 that shares a side with the first square. Next, add a 2-by-2 square, with one side along the sides of
both
of the first two squares. Next, add a square with one side that goes along the sides of the previous two squares you created. Next, do it again.
Pause here for your teacher to check your work.
Write a sequence that lists the side lengths of the squares you drew.
Predict the next two terms in the sequence and draw the corresponding squares to check your predictions.
Describe how each square’s side length depends on previous side lengths.
Let
\(F(n)\)
be the side length of the
\(n\)
th
square. So
\(F(1)=1\)
and
\(F(2)=1\)
. Write a recursive definition for
\(F\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Is the Fibonacci sequence arithmetic, geometric, or neither? Explain how you know.
Look at quotients
\(\frac{F(1)}{F(0)}, \frac{F(2)}{F(1)}, \frac{F(3)}{F(2)}, \frac{F(4)}{F(3)}, \frac{F(5)}{F(4)}\)
. What do you notice about this sequence of numbers?
The 15th through 19th Fibonacci numbers are 610, 987, 1597, 2584, 4181. What do you notice about the quotients
\(\frac{F(16)}{F(15)}, \frac{F(17)}{F(16)}, \frac{F(18)}{F(17)}, \frac{F(19)}{F(18)}\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may not realize that it is okay to use both
\(F(n-1)\)
and
\(F(n-2)\)
when defining a function recursively. Encourage these students to think about how the “previous side lengths” for term
\(F(n)\)
can be expressed using function notation.
Activity Synthesis
Display the sequence 1, 1, 2, 3, 5, 8, 13 and the recursive definition
\(F(1)=1, F(2)=1, \)
\(F(n)=F(n-1)+F(n-2)\)
,
\(n\ge3\)
for all to see and check their answer against. Either give students a minute to think about how to write a definition for the
\(n^{\text{th}}\)
term of the sequence, or just tell them that it is more complicated to derive than class time allows. An important modeling point to make is that the goal is to create a model that suits the needs of what you are trying to do. If we wanted to know the value of
\(F(100)\)
, then a spreadsheet and a recursive definition is likely going to be faster than working out a definition for the
\(n^{\text{th}}\)
term.
Ask students, "What is the domain of this function? Does it have any restrictions?" (The integers
\(n\)
where
\(n\ge1\)
or equivalent. We can always add another square onto the image, even if only in our imaginations once the squares get too big.) Select students to share their responses.
If time allows, tell students that we could define the Fibonacci sequence as
\(F(0)=1, F(1)=1, \)
\(F(n)=F(n-1)+F(n-2)\)
,
\(n\ge2\)
and we would still get the same list of numbers.


correct_option_index: 5

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